매개변수와 극좌표 전체 문제

For the following exercises, find ${d}^{2}y/d{x}^{2}.$

$\begin{array}{cc}x={e}^{-t}, & y=t\end{array}{e}^{2t}$

For the following exercises, find points on the curve at which tangent line is horizontal or vertical.

$\begin{array}{cc}x=t({t}^{2}-3), & y=3({t}^{2}-3)\end{array}$

For the following exercises, find points on the curve at which tangent line is horizontal or vertical.

$\begin{array}{cc}x=\frac{3t}{1+{t}^{3}}, & y=\frac{3{t}^{2}}{1+{t}^{3}}\end{array}$

For the following exercises, find $dy/dx$ at the value of the parameter.

$\begin{array}{cc}x=\sqrt{t}, & y=2t+4,\quad t=9\end{array}$

For the following exercises, find $dy/dx$ at the value of the parameter.

$\begin{array}{cc}x=4\,\cos (2\pi s), & y=3\,\sin \end{array}(2\pi s),\quad s=-\frac{1}{4}$

For the following exercises, find ${d}^{2}y/d{x}^{2}$ at the given point without eliminating the parameter.

$\begin{array}{ccc}x=\frac{1}{2}{t}^{2}, & y=\frac{1}{3}{t}^{3}, & t=2\end{array}$

For the following exercises, find ${d}^{2}y/d{x}^{2}$ at the given point without eliminating the parameter.

$x=\sqrt{t},\quad y=2t+4,\quad t=1$

Find t intervals on which the curve $x=3{t}^{2},y={t}^{3}-t$ is concave up as well as concave down.

For the following exercises, find ${d}^{2}y/d{x}^{2}$ at the given point without eliminating the parameter.

Determine the concavity of the curve $x=2t+\ln \,t,y=2t-\ln \,t.$

Find the area bounded by the curve $x=\cos \,t,y={e}^{t},0\le t\le \frac{\pi }{2}$ and the lines $y=1$ and $x=0.$

For the following exercises, find ${d}^{2}y/d{x}^{2}$ at the given point without eliminating the parameter.

Find the area enclosed by the ellipse $x=a\,\cos \,\theta ,y=b\,\sin \,\theta ,0\le \theta <2\pi .$

For the following exercises, find the arc length of the curve on the indicated interval of the parameter.

$x=4t+3,\quad y=3t-2,\quad 0\le t\le 2$

For the following exercises, find the arc length of the curve on the indicated interval of the parameter.

$\begin{array}{ccc}x=\frac{1}{3}{t}^{3}, & y=\frac{1}{2}{t}^{2}, & 0\le t\le 1\end{array}$

For the following exercises, find the arc length of the curve on the indicated interval of the parameter.

$\begin{array}{ccc}x=\cos (2t), & y=\sin (2t), & 0\le t\le \frac{\pi }{2}\end{array}$

For the following exercises, find the arc length of the curve on the indicated interval of the parameter.

$\begin{array}{ccc}x=1+{t}^{2}, & y={(1+t)}^{3}, & 0\le t\le 1\end{array}$

$\begin{array}{ccc}x={e}^{t}\cos \,t, & y={e}^{t}\sin \,t, & 0\le t\le \frac{\pi }{2}\end{array}$ (Use a CAS for this and express the answer as a decimal rounded to three places.)

For the following exercises, find the arc length of the curve on the indicated interval of the parameter.

$x=a\,{\cos }^{3}\theta ,y=a\,{\sin }^{3}\theta$ on the interval $[0,2\pi )$ (the hypocycloid)

Find the length of one arch of the cycloid $x=4(t-\sin \,t),y=4(1-\cos \,t).$

Find the distance traveled by a particle with position $(x,y)$ as t varies in the given time interval: $\begin{array}{ccc}x={\sin }^{2}t, & y={\cos }^{2}t, & 0\le t\le 3\pi \end{array}.$

Find the length of one arch of the cycloid $x=\theta -\sin \,\theta ,y=1-\cos \,\theta .$